The length of Jordan algebras and beyond

We prove that the length of the commutative Jordan algebras over a field of the characteristic different from 2 is bounded by the dimension from above. This bound is the same as for the class of associative algebras, but we demonstrate that the length of a given associative algebra can be either greater or lesser or equal to the length of the corresponding adjoint Jordan algebra. We also show that the Jordan identity by itself (or even with commutativity in characteristic 2) does not guarantee a linear bound on growth. In addition, we compute the exact length of bicomplex numbers and biquaternions.